# If the hypotenuse of a triangle is 12 cm and

If the hypotenuse of a triangle is 12 cm and the measure of its legs are in the ratio $1:2$, then what are the measures of it’s legs?
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bedevijuo3e

Step 1
Here is a way to solve without directly using Pythagoras theorem.
Let a, b and c be the side lengths of the given right triangle such that c is is the length of hypotenuse.
Measure of legs are in the ratio $1:2$
Let
Let R and r be the circumradius and inradius of right triangle respectively.
$R=\frac{c}{2}=\frac{12}{2}=6$
$r=\frac{ab}{a+b+c}=\frac{x×2x}{x+2x+12}=\frac{2{x}^{2}}{3x+12}$
We have,
$R+r=\frac{a+b}{2}$
$6+\frac{2{x}^{2}}{3x+12}=\frac{x+2x}{2}$
$\left(3x+12\right)\left(3x-12\right)=4{x}^{2}$
$5{x}^{2}-144=0⇒x=\frac{12}{\sqrt{5}}$
$a=\frac{12}{\sqrt{5}}$
$b=2×\frac{12}{\sqrt{5}}=\frac{24}{\sqrt{5}}$

###### Not exactly what you’re looking for?
junoon363km
Step 1
If the triangle has a hypotenuse, I’ll assume that the triangle is right angled.
Mr. Pythagoras says that the hypotenuse squared = leg 1 squared plus leg 2 squared.

${\left(1x\right)}^{2}+{\left(2x\right)}^{2}={12}^{2}$
$\left(1{x}^{2}\right\}\right)+\left(4{x}^{2}\right)=144$
$5{x}^{2}={12}^{2}$
${x}^{2}=\frac{144}{5}$
${x}^{2}=28.8$

Step 2
Proof:
If both sides of this equation are equal, then the answer is right.
${12}^{2}=\ne {5.367}^{2}+{10.733}^{2}$
$144=\ne 28.8+115.2$
$144=144$
One leg of the triangle is 5.367 cm and the second leg is 10.733 cm.